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A022103 Fibonacci sequence beginning 1, 13. 4
1, 13, 14, 27, 41, 68, 109, 177, 286, 463, 749, 1212, 1961, 3173, 5134, 8307, 13441, 21748, 35189, 56937, 92126, 149063, 241189, 390252, 631441, 1021693, 1653134, 2674827, 4327961, 7002788, 11330749, 18333537, 29664286, 47997823, 77662109, 125659932, 203322041, 328981973 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(13;n-1-k,k) for n>=1, a(-1)=12. These are the SW-NE diagonals in P(13;n,k), the (13,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

In general, for b Fibonacci sequence beginning with 1, h, we have:

b(n) = (2^(-1-n)*((1 - sqrt(5))^n*(1 + sqrt(5) - 2*h) + (1 + sqrt(5))^n*(-1 + sqrt(5) + 2*h)))/sqrt(5). - Herbert Kociemba, Dec 18 2011

Pisano period lengths: 1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, ... (is this A106291?). - R. J. Mathar, Aug 10 2012

LINKS

Table of n, a(n) for n=0..37.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (1,1).

FORMULA

a(n) = a(n-1) + a(n-2) for n>=2, a(0)=1, a(1)=13, and a(-1):=12.

G.f.: (1 + 12*x)/(1 - x - x^2).

a(n) = ((1 + sqrt(5))^n-(1 - sqrt(5))^n)/(2^n*sqrt(5))+ 6*((1 + sqrt(5))^(n-1)-(1 - sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)) for n>0. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009

a(n) = 12*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012

a(n) = 13*A000045(n) + A000045(n-1). - Paolo P. Lava, May 19 2015

a(n) = 14*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017

a(n) = Lucas(n+5) - 5*Lucas(n). - Bruno Berselli, Dec 30 2016

MAPLE

with(numtheory): with(combinat): P:=proc(q) local n;

for n from 0 to q do print(13*fibonacci(n)+fibonacci(n-1));

od; end: P(30); # Paolo P. Lava, May 19 2015

MATHEMATICA

LinearRecurrence[{1, 1}, {1, 13}, 40] (* or *) Table[LucasL[n + 5] - 5 LucasL[n], {n, 0, 40}] (* Bruno Berselli, Dec 30 2016 *)

PROG

(MAGMA) a0:=1; a1:=13; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

CROSSREFS

a(n) = A109754(12, n+1) = A101220(12, 0, n+1).

Cf. A000032, A000045.

Sequence in context: A022803 A112653 A015905 * A224224 A041342 A041344

Adjacent sequences:  A022100 A022101 A022102 * A022104 A022105 A022106

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 21 23:43 EDT 2017. Contains 290940 sequences.